Var estimation with using Garch family
Elgun Ismayilov
2024-06-11
Importing necessary libraries
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## volatilityLets load additional function defined which will be used to to easily compare information criteria for ARCH and GARCH models.
source("/Users/elgun/Desktop/ESTIMATING-VALUE-AT-RISK-OF-A-PORTFOLIO-WITH-GARCH-FAMILY-MODELS-Time-Series-project/functions/compare_ICs_ugarchfit.R")Import the data
# Fetch data for Dow Jones
Dow_Jones <- getSymbols("^DJI", src = "yahoo", from = "2019-01-01", to = "2024-06-01", auto.assign = FALSE)
# Rename the column
colnames(Dow_Jones) <- c("Open", "High", "Low", "Close", "Volume", "Adjusted")
# Fetch data for TSMC
Tmsc <- getSymbols("TSM", src = "yahoo", from = "2019-01-01", to = "2024-06-01", auto.assign = FALSE)
# Rename the column
colnames(Tmsc) <- c("Open", "High", "Low", "Close", "Volume", "Adjusted")
# Fetch data for Pound
Pound <- getSymbols("GBPUSD=X", src = "yahoo", from = "2019-01-01", to = "2024-06-01", auto.assign = FALSE)## Warning: GBPUSD=X contains missing values. Some functions will not work if
## objects contain missing values in the middle of the series. Consider using
## na.omit(), na.approx(), na.fill(), etc to remove or replace them.# Rename the column
colnames(Pound) <- c("Open", "High", "Low", "Close", "Volume","Adjusted")
# Fetch data for Silver
Silver <- getSymbols("SI=F", src = "yahoo", from = "2019-01-01", to = "2024-06-01", auto.assign = FALSE)## Warning: SI=F contains missing values. Some functions will not work if objects
## contain missing values in the middle of the series. Consider using na.omit(),
## na.approx(), na.fill(), etc to remove or replace them.# Rename the column
colnames(Silver) <- c("Open", "High", "Low", "Close", "Volume","Adjusted")
# Fetch data for BNB
BNB <- getSymbols("BNB-USD", src = "yahoo", from = "2019-01-01", to = "2024-06-01", auto.assign = FALSE)
# Rename the column
colnames(BNB) <- c("Open", "High", "Low", "Close", "Volume", "Adjusted")
print(Pound)## Open High Low Close Volume Adjusted
## 2019-01-01 1.275022 1.275673 1.271456 1.273804 0 1.273804
## 2019-01-02 1.275234 1.277335 1.258463 1.275429 0 1.275429
## 2019-01-03 1.251878 1.263823 1.251878 1.252191 0 1.252191
## 2019-01-04 1.262897 1.273966 1.261782 1.262881 0 1.262881
## 2019-01-07 1.273285 1.278772 1.272426 1.273496 0 1.273496
## 2019-01-08 1.278691 1.279230 1.271488 1.278609 0 1.278609
## 2019-01-09 1.273885 1.280295 1.271779 1.273788 0 1.273788
## 2019-01-10 1.279951 1.280115 1.273058 1.279967 0 1.279967
## 2019-01-11 1.275836 1.283746 1.271666 1.275396 0 1.275396
## 2019-01-14 1.284984 1.292775 1.282101 1.284984 0 1.284984
## ...
## 2024-05-19 1.270793 1.271294 1.269148 1.270648 0 1.270648
## 2024-05-20 1.271052 1.272734 1.269293 1.271084 0 1.271084
## 2024-05-21 1.271036 1.276129 1.270099 1.271133 0 1.271133
## 2024-05-22 1.272087 1.274697 1.270616 1.272103 0 1.272103
## 2024-05-23 1.269374 1.275136 1.268279 1.269406 0 1.269406
## 2024-05-26 1.273447 1.277900 1.273447 1.273610 0 1.273610
## 2024-05-27 1.277270 1.279984 1.276308 1.277237 0 1.277237
## 2024-05-28 1.276000 1.277172 1.270713 1.275836 0 1.275836
## 2024-05-29 1.269825 1.274762 1.268183 1.269799 0 1.269799
## 2024-05-30 1.273026 1.276552 1.270164 1.273042 0 1.273042Handling with NA values use na_locf
Dow_Jones$Adjusted <- na_locf(Dow_Jones$Adjusted)
Tmsc$Adjusted <- na_locf(Tmsc$Adjusted)
Pound$Adjusted <- na_locf(Pound$Adjusted)
Silver$Adjusted <- na_locf(Silver$Adjusted)
BNB$Adjusted <- na_locf(BNB$Adjusted)Log return of ecah asset
Dow_Jones$r <- diff.xts(log(Dow_Jones$Adjusted))
Tmsc$r <- diff.xts(log(Tmsc$Adjusted))
Pound$r <- diff.xts(log(Pound$Adjusted))
Silver$r <- diff.xts(log(Silver$Adjusted))
BNB$r <- diff.xts(log(BNB$Adjusted))
print(Dow_Jones$r)## r
## 2019-01-02 NA
## 2019-01-03 -0.0286782307
## 2019-01-04 0.0323943972
## 2019-01-07 0.0041814381
## 2019-01-08 0.0108245398
## 2019-01-09 0.0038463031
## 2019-01-10 0.0051294230
## 2019-01-11 -0.0002487903
## 2019-01-14 -0.0035949504
## 2019-01-15 0.0064929297
## ...
## 2024-05-17 0.0033606129
## 2024-05-20 -0.0049322096
## 2024-05-21 0.0016621227
## 2024-05-22 -0.0050776824
## 2024-05-23 -0.0153878018
## 2024-05-24 0.0001107860
## 2024-05-28 -0.0055627365
## 2024-05-29 -0.0106430531
## 2024-05-30 -0.0086230600
## 2024-05-31 0.0149704959Portfolio construction
# Calculate the combined returns of all assets
combined_returns <- Dow_Jones$r + Tmsc$r + Pound$r + Silver$r + BNB$r
# Calculate the portfolio returns with equal weights
weight <- 0.2
portfolio_returns <- combined_returns * weight
# Convert the xts object to a data frame
portfolio_df1 <- data.frame(date = index(combined_returns), portfolio_returns = coredata(portfolio_returns))
portfolio_df1 <- na.omit(portfolio_df1)
# Ensure the date column is in Date format
portfolio_df1$date <- as.Date(portfolio_df1$date)
# Removing observations after `2023-01-01`
portfolio_df <- portfolio_df1 %>%
filter(date <= as.Date("2023-01-01"))
# Print the portfolio data frame
print(na.omit(portfolio_df))## date r
## 1 2019-01-03 -2.893906e-02
## 2 2019-01-04 1.695959e-02
## 3 2019-01-07 3.633670e-04
## 4 2019-01-08 1.155042e-02
## 5 2019-01-09 4.202983e-03
## 6 2019-01-10 -1.782659e-02
## 7 2019-01-11 5.799201e-03
## 8 2019-01-14 1.664867e-02
## 9 2019-01-15 -5.516295e-03
## 10 2019-01-16 7.951595e-03
## 11 2019-01-17 1.127519e-02
## 12 2019-01-18 9.962925e-03
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## 762 2022-05-25 2.307512e-03
## 763 2022-05-26 -7.448659e-03
## 764 2022-05-31 -3.245514e-03
## 765 2022-06-01 -1.497179e-02
## 766 2022-06-02 1.584341e-02
## 767 2022-06-06 -1.523901e-03
## 768 2022-06-07 1.495335e-03
## 769 2022-06-08 -5.128958e-03
## 770 2022-06-09 -1.198235e-02
## 771 2022-06-13 -4.926194e-02
## 772 2022-06-14 -1.592440e-03
## 773 2022-06-15 2.238474e-02
## 774 2022-06-16 -2.878958e-02
## 775 2022-06-21 1.304741e-02
## 776 2022-06-22 -1.335989e-02
## 777 2022-06-23 9.292173e-03
## 778 2022-06-27 -5.502174e-04
## 779 2022-06-28 -1.728528e-02
## 780 2022-06-29 -1.016152e-02
## 781 2022-06-30 -9.515992e-03
## 782 2022-07-05 -1.217776e-02
## 783 2022-07-06 5.435152e-03
## 784 2022-07-07 1.993624e-02
## 785 2022-07-11 -1.765179e-02
## 786 2022-07-12 -8.258522e-03
## 787 2022-07-13 1.531388e-02
## 788 2022-07-14 5.635317e-04
## 789 2022-07-18 1.067603e-02
## 790 2022-07-19 1.343173e-02
## 791 2022-07-20 -7.564650e-03
## 792 2022-07-21 1.019851e-02
## 793 2022-07-25 -1.469257e-02
## 794 2022-07-26 -9.591014e-04
## 795 2022-07-27 3.032817e-02
## 796 2022-07-28 2.194918e-02
## 797 2022-08-01 -2.217445e-03
## 798 2022-08-02 -6.842366e-03
## 799 2022-08-03 1.112988e-02
## 800 2022-08-04 1.463117e-02
## 801 2022-08-08 5.288491e-03
## 802 2022-08-09 -8.072380e-03
## 803 2022-08-10 1.818949e-02
## 804 2022-08-11 -4.889635e-03
## 805 2022-08-15 -1.695455e-03
## 806 2022-08-16 -5.501227e-03
## 807 2022-08-17 -1.340526e-02
## 808 2022-08-18 -9.913651e-03
## 809 2022-08-22 -1.221929e-02
## 810 2022-08-23 2.482007e-03
## 811 2022-08-24 -4.395936e-03
## 812 2022-08-25 1.284487e-02
## 813 2022-08-29 -7.214819e-05
## 814 2022-08-30 -1.084154e-02
## 815 2022-08-31 -7.458912e-03
## 816 2022-09-01 -7.056688e-03
## 817 2022-09-06 -1.440190e-02
## 818 2022-09-07 1.841843e-02
## 819 2022-09-08 5.391081e-03
## 820 2022-09-12 1.455110e-02
## 821 2022-09-13 -3.450432e-02
## 822 2022-09-14 5.145130e-03
## 823 2022-09-15 -1.515782e-02
## 824 2022-09-19 6.361820e-03
## 825 2022-09-20 -9.350892e-03
## 826 2022-09-21 -6.214923e-03
## 827 2022-09-22 4.817024e-03
## 828 2022-09-26 -8.351628e-03
## 829 2022-09-27 -5.847934e-03
## 830 2022-09-28 1.585357e-02
## 831 2022-09-29 -5.670249e-03
## 832 2022-10-03 2.833022e-02
## 833 2022-10-04 2.894550e-02
## 834 2022-10-05 -4.268315e-03
## 835 2022-10-06 -1.012693e-02
## 836 2022-10-10 -1.839002e-02
## 837 2022-10-11 -1.556318e-02
## 838 2022-10-12 -1.656515e-03
## 839 2022-10-13 1.724759e-02
## 840 2022-10-17 1.800982e-02
## 841 2022-10-18 -4.395119e-03
## 842 2022-10-19 -6.628747e-03
## 843 2022-10-20 9.053478e-04
## 844 2022-10-24 -5.963001e-03
## 845 2022-10-25 1.355693e-02
## 846 2022-10-26 7.667620e-03
## 847 2022-10-27 -2.135221e-03
## 848 2022-10-31 5.889987e-03
## 849 2022-11-01 1.948747e-03
## 850 2022-11-02 -9.712056e-03
## 851 2022-11-03 6.608405e-04
## 852 2022-11-04 3.379665e-02
## 853 2022-11-07 5.975885e-03
## 854 2022-11-08 1.221613e-02
## 855 2022-11-09 -4.614741e-02
## 856 2022-11-10 5.089026e-02
## 857 2022-11-11 4.248730e-03
## 858 2022-11-14 3.216197e-03
## 859 2022-11-15 1.288259e-02
## 860 2022-11-16 -4.039468e-03
## 861 2022-11-17 -2.674990e-03
## 862 2022-11-18 6.823952e-03
## 863 2022-11-21 -1.458182e-02
## 864 2022-11-22 1.889270e-02
## 865 2022-11-23 2.572649e-02
## 866 2022-11-25 8.929453e-04
## 867 2022-11-28 -2.344045e-02
## 868 2022-11-29 2.853585e-03
## 869 2022-11-30 1.987077e-02
## 870 2022-12-01 4.239908e-03
## 871 2022-12-02 4.859472e-03
## 872 2022-12-05 -1.195771e-02
## 873 2022-12-06 -8.288218e-03
## 874 2022-12-07 -7.619792e-04
## 875 2022-12-08 1.372941e-02
## 876 2022-12-09 -7.331250e-04
## 877 2022-12-12 -5.943724e-03
## 878 2022-12-13 3.110392e-03
## 879 2022-12-14 -2.203204e-03
## 880 2022-12-15 -2.304064e-02
## 881 2022-12-16 -3.077239e-02
## 882 2022-12-19 -1.158869e-02
## 883 2022-12-20 1.723182e-02
## 884 2022-12-21 2.682055e-03
## 885 2022-12-22 -1.365992e-02
## 886 2022-12-23 2.053295e-03
## 887 2022-12-27 3.362720e-03
## 888 2022-12-28 -1.173805e-02
## 889 2022-12-29 1.538127e-02
## 890 2022-12-30 -6.366028e-03Log-returns plot
plot(portfolio_df$r,
col = "red",
major.ticks = "years",
grid.ticks.on = "years",
main = "Log-returns of Portfolio")## Warning in plot.window(...): "major.ticks" is not a graphical parameter## Warning in plot.window(...): "grid.ticks.on" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.ticks" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "grid.ticks.on" is not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.ticks" is
## not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "grid.ticks.on" is
## not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.ticks" is
## not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "grid.ticks.on" is
## not a graphical parameter## Warning in box(...): "major.ticks" is not a graphical parameter## Warning in box(...): "grid.ticks.on" is not a graphical parameter## Warning in title(...): "major.ticks" is not a graphical parameter## Warning in title(...): "grid.ticks.on" is not a graphical parameter
Plot of ACF for returns
acf(portfolio_df$r,lag.max = 36, na.action = na.pass,col="darkblue",lwd=7,main="ACF of Portfolio returns")
acf(portfolio_df$r, lag.max = 36, na.action = na.pass, ylim = c(-0.2, 0.2),
col = "darkblue", lwd = 7, main = "ACF of Portfolio returns")
Plot of ACF for squared returns.
acf(portfolio_df$r^2,lag.max = 100,na.action = na.pass,col="darkblue",lwd=7,main="ACF of Portfolio squared returns")
acf(portfolio_df$r^2,lag.max = 100,na.action = na.pass,
ylim= c(0, 0.5),col="darkblue",lwd=7,main="CF of Portfolio squared returns")
Lag 1,2,3,4,6,8 seems significant. ## Normality test
basicStats((portfolio_df$r))## X..portfolio_df.r
## nobs 890.000000
## NAs 0.000000
## Minimum -0.154560
## Maximum 0.111584
## 1. Quartile -0.007075
## 3. Quartile 0.009494
## Mean 0.001027
## Median 0.001057
## Sum 0.913659
## SE Mean 0.000567
## LCL Mean -0.000086
## UCL Mean 0.002139
## Variance 0.000286
## Stdev 0.016908
## Skewness -0.897962
## Kurtosis 12.613657Skewness is negative and we also observe strong excess kurtosis.
tibble(r = as.numeric(portfolio_df$r)) %>%
ggplot(aes(r)) +
geom_histogram(aes(y =..density..),
colour = "black",
fill = "pink") +
stat_function(fun = dnorm,
args = list(mean = mean(portfolio_df$r),
sd = sd(portfolio_df$r))) +
theme_bw() +
labs(
title = "Density of the portfolio log-returns",
y = "", x = "",
caption = "source: own calculations"
)## Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
## ℹ Please use `after_stat(density)` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
As we can see, the distribution of the returns is highly
leptokurtic.
jarque.bera.test(portfolio_df$r)##
## Jarque Bera Test
##
## data: portfolio_df$r
## X-squared = 6053.1, df = 2, p-value < 2.2e-16The null hypothesis about normality strongly rejected!
Testing for ARCH effects.
ArchTest(portfolio_df$r,lags = 5)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: portfolio_df$r
## Chi-squared = 117.13, df = 5, p-value < 2.2e-16There is very strong evidence to reject the null-hypothesis that there are no ARCH effects in the data.
Standardization of returns.
portfolio_df$stan <-(portfolio_df$r-mean(portfolio_df$r, na.rm=T))/ sd(portfolio_df$r, na.rm = T)
tail(portfolio_df$stan)## [1] -0.86859005 0.06072186 0.13816399 -0.75492673 0.84896594 -0.43721426Histogram of standart return
hist(portfolio_df$stan, prob=T, breaks = 40)
curve(dnorm(x, mean(portfolio_df$stan, na.rm= T),
sd=sd(portfolio_df$stan, na.rm = T)),
col="darkblue",lwd=2, add=TRUE)
Descriptive statistics of standardized returns .
basicStats(portfolio_df$stan)## X..portfolio_df.stan
## nobs 890.000000
## NAs 0.000000
## Minimum -9.201715
## Maximum 6.538615
## 1. Quartile -0.479149
## 3. Quartile 0.500763
## Mean 0.000000
## Median 0.001793
## Sum 0.000000
## SE Mean 0.033520
## LCL Mean -0.065788
## UCL Mean 0.065788
## Variance 1.000000
## Stdev 1.000000
## Skewness -0.897962
## Kurtosis 12.6136571% empirical quantile
quantile(portfolio_df$stan,0.01, na.rm=T)## 1%
## -2.631463For comparison: 1% quantile of standard normal distribution
qnorm(0.01,0,1)## [1] -2.326348Build GARCH models
Estimating the GARCH(1,1)
Here, first we have to define a model specification:
spec <- ugarchspec(variance.model = list(model="sGARCH",garchOrder= c(1,1)),mean.model = list(armaOrder = c(0,0), include.mean= T),distribution.model = "norm")Estimate the model:
portfolio_df.garch11 <- ugarchfit(spec = spec,
data = portfolio_df$r)Now, we can see the results:
portfolio_df.garch11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000785 0.000451 1.7381 0.082189
## omega 0.000014 0.000003 5.5947 0.000000
## alpha1 0.146300 0.019368 7.5537 0.000000
## beta1 0.809070 0.017357 46.6137 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000785 0.000446 1.7576 0.078818
## omega 0.000014 0.000008 1.9153 0.055453
## alpha1 0.146300 0.027753 5.2716 0.000000
## beta1 0.809070 0.056667 14.2777 0.000000
##
## LogLikelihood : 2473.557
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.5496
## Bayes -5.5280
## Shibata -5.5496
## Hannan-Quinn -5.5413
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.756 0.1852
## Lag[2*(p+q)+(p+q)-1][2] 1.943 0.2724
## Lag[4*(p+q)+(p+q)-1][5] 2.782 0.4483
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2575 0.6118
## Lag[2*(p+q)+(p+q)-1][5] 1.2208 0.8080
## Lag[4*(p+q)+(p+q)-1][9] 1.9909 0.9061
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.0198 0.500 2.000 0.8881
## ARCH Lag[5] 0.8020 1.440 1.667 0.7925
## ARCH Lag[7] 1.1231 2.315 1.543 0.8927
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.6581
## Individual Statistics:
## mu 0.28254
## omega 0.89762
## alpha1 0.07765
## beta1 0.12538
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.2984 0.1945
## Negative Sign Bias 0.1061 0.9155
## Positive Sign Bias 0.1665 0.8678
## Joint Effect 2.5933 0.4587
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 37.19 0.0075096
## 2 30 54.16 0.0031032
## 3 40 64.79 0.0058664
## 4 50 87.42 0.0006097
##
##
## Elapsed time : 0.03735805Interpration of results
1.Ljung-Box Tests for Residuals and Squared Residuals: The Ljung-Box tests for both standardized residuals and squared standardized residuals indicate that there is no significant autocorrelation remaining in the residuals at the tested lags.
2.Arch Test for Residuals: No remaining ARCH effects, suggesting the model has adequately captured conditional heteroscedasticity.
3.Nyblom Stability Test: Indicates potential instability in the model parameters over time, particularly the constant term (ω).
4.Sign Bias Test: Shows no evidence of asymmetry in the impact of positive and negative shocks on volatility, suggesting the model handles these shocks symmetrically.
5.Adjusted Pearson Goodness-of-Fit Test: Reveals significant deviations from normality in the model’s residuals, indicating the model may not adequately capture the true distribution of the data.
Conditional SD (vs |returns|)
plot(portfolio_df.garch11, which=3)
There are periods where the volatility is relatively low and stable, suggesting less market turbulence.
ACF of Standardized Residuals
plot(portfolio_df.garch11, which=10)
ACF of Squared Standardized Residuals
plot(portfolio_df.garch11, which=11)
plot(portfolio_df.garch11, which=12)
Positive and negative shocks of equal magnitude have the same impact on volatility. The U-shaped curve indicates that both positive and negative shocks increase volatility, but the impact diminishes as the magnitude of the shock decreases.
The EGARCH model
Here, first we have to define a model specification:
spec <- ugarchspec(# variance equation
variance.model = list(model = "eGARCH",
garchOrder = c(1, 1)),
# sGARCH would stand for standard GARCH model
# mean equation
mean.model = list(armaOrder = c(0, 0),
include.mean = TRUE),
# assumed distribution of errors
distribution.model = "norm")Estimate the model:
portfolio_df.egarch11 <- ugarchfit(spec = spec,
data = portfolio_df$r)Now, we can see the results:
portfolio_df.egarch11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.001059 0.000308 3.43698 0.000588
## omega -0.375593 0.109973 -3.41531 0.000637
## alpha1 -0.010965 0.021057 -0.52075 0.602540
## beta1 0.953260 0.013109 72.71627 0.000000
## gamma1 0.264774 0.033653 7.86774 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.001059 0.000211 5.00693 0.000001
## omega -0.375593 0.119341 -3.14722 0.001648
## alpha1 -0.010965 0.039602 -0.27689 0.781863
## beta1 0.953260 0.013766 69.24699 0.000000
## gamma1 0.264774 0.069090 3.83230 0.000127
##
## LogLikelihood : 2473.587
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.5474
## Bayes -5.5205
## Shibata -5.5474
## Hannan-Quinn -5.5371
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.482 0.2234
## Lag[2*(p+q)+(p+q)-1][2] 1.872 0.2849
## Lag[4*(p+q)+(p+q)-1][5] 2.879 0.4295
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1083 0.7421
## Lag[2*(p+q)+(p+q)-1][5] 2.2163 0.5683
## Lag[4*(p+q)+(p+q)-1][9] 3.0676 0.7478
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.04129 0.500 2.000 0.8390
## ARCH Lag[5] 0.76302 1.440 1.667 0.8043
## ARCH Lag[7] 0.99522 2.315 1.543 0.9143
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.7094
## Individual Statistics:
## mu 0.33897
## omega 0.10151
## alpha1 0.07545
## beta1 0.10189
## gamma1 0.06438
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.5077 0.1320
## Negative Sign Bias 0.1736 0.8623
## Positive Sign Bias 0.5713 0.5680
## Joint Effect 2.7772 0.4273
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 39.62 0.003672
## 2 30 45.46 0.026528
## 3 40 64.61 0.006114
## 4 50 78.99 0.004244
##
##
## Elapsed time : 0.04063892Interpration
1.Ljung-Box Tests for Residuals and Squared Residuals: The Ljung-Box test results indicate no significant serial correlation, suggesting that the model is appropriate for capturing the autocorrelation in the time series data.Similarly, the test results for the squared residuals indicate no significant serial correlation, suggesting that the model adequately captures the conditional heteroscedasticity (volatility clustering).
2.Arch Test for Residuals: The fitted model adequately captures both autocorrelation and conditional heteroscedasticity in the time series data, indicating no significant serial correlation or ARCH effects in the residuals.
3.Nyblom Stability Test: The Nyblom stability test indicates that all parameters in the model are stable, with no significant evidence of instability in either individual or joint tests.
4.Sign Bias Test: Based on the Sign Bias Test results, there is no significant indication of a systematic bias in the signs of the residuals, suggesting that the model’s residuals exhibit no discernible pattern in their signs.
5.Adjusted Pearson Goodness-of-Fit Test: The Adjusted Pearson Goodness-of-Fit Test results suggest that the model does not adequately fit the observed data across all groups. The low p-values indicate significant evidence against the model’s fit, implying that the model may need adjustments or improvements to better capture the characteristics of the data.
Conditional SD (vs |returns|)
plot(portfolio_df.egarch11, which = 3)
There are periods where the volatility is relatively low and stable, suggesting less market turbulence.
ACF of Standardized Residuals
plot(portfolio_df.egarch11, which = 10)
ACF of Squared Standardized Residuals
plot(portfolio_df.egarch11, which = 11)
News-impact curve
plot(portfolio_df.egarch11, which = 12)
Positive and negative shocks of equal magnitude have the same impact on volatility. The U-shaped curve indicates that both positive and negative shocks increase volatility, but the impact diminishes as the magnitude of the shock decreases.
The GARCH-in-Mean model
Let’s first define a model specification:
spec <- ugarchspec(# variance equation
variance.model = list(model = "sGARCH",
# sGARCH = standard GARCH
garchOrder = c(1, 1)),
# mean equation - lets turn on the intercept term
mean.model = list(armaOrder = c(0, 0),
include.mean = TRUE,
# we add an element to the mean equation,
# which can be either stdev (archpow 1)
# or var (archpow=2)
archm = TRUE, archpow = 1),
# assumed distribution of errors
distribution.model = "norm")Then, we can estimate the model:
portfolio_df.garchm11 <- ugarchfit(spec = spec,
data = portfolio_df$r)Let’s examine the results:
portfolio_df.garchm11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.001444 0.001847 -0.78193 0.434253
## archm 0.162857 0.130265 1.25020 0.211227
## omega 0.000015 0.000003 4.81129 0.000001
## alpha1 0.149075 0.019749 7.54855 0.000000
## beta1 0.805595 0.016655 48.37068 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.001444 0.001685 -0.85678 0.39157
## archm 0.162857 0.114424 1.42328 0.15465
## omega 0.000015 0.000010 1.53070 0.12584
## alpha1 0.149075 0.028109 5.30356 0.00000
## beta1 0.805595 0.066298 12.15106 0.00000
##
## LogLikelihood : 2474.321
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.5490
## Bayes -5.5221
## Shibata -5.5491
## Hannan-Quinn -5.5387
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.885 0.1697
## Lag[2*(p+q)+(p+q)-1][2] 2.094 0.2477
## Lag[4*(p+q)+(p+q)-1][5] 2.899 0.4257
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2724 0.6018
## Lag[2*(p+q)+(p+q)-1][5] 1.3477 0.7773
## Lag[4*(p+q)+(p+q)-1][9] 2.1878 0.8814
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.02703 0.500 2.000 0.8694
## ARCH Lag[5] 0.86152 1.440 1.667 0.7744
## ARCH Lag[7] 1.25066 2.315 1.543 0.8697
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.2369
## Individual Statistics:
## mu 0.36461
## archm 0.36412
## omega 0.85120
## alpha1 0.08096
## beta1 0.11809
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.910554 0.3628
## Negative Sign Bias 0.035967 0.9713
## Positive Sign Bias 0.009599 0.9923
## Joint Effect 1.504741 0.6812
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 36.61 0.008883
## 2 30 51.73 0.005841
## 3 40 74.94 0.000472
## 4 50 84.94 0.001101
##
##
## Elapsed time : 0.131501Interpration
1.Ljung-Box Tests for Residuals and Squared Residuals: Both the Weighted Ljung-Box tests on standardized residuals and standardized squared residuals suggest that there is no evidence of serial correlation in the residuals at the 5% significance level.
2.Arch Test for Residuals: These results suggest that there is no significant evidence of autoregressive conditional heteroskedasticity (ARCH effects) in the residuals of the model at the 5% significance level.
3.Nyblom Stability Test: he Nyblom stability test suggests that the model’s parameters are stable.
4.Sign Bias Test: the Sign Bias Test suggests that there’s no significant systematic bias in the signs of the residuals.
5.Adjusted Pearson Goodness-of-Fit Test: Reject the null hypothesis and conclude that there is a significant difference between the observed and expected frequencies in each group.
Conditional SD (vs |returns|)
plot(portfolio_df.garchm11, which = 3)
There are periods where the volatility is relatively low and stable, suggesting less market turbulence.
ACF of Standardized Residuals
plot(portfolio_df.garchm11, which = 10)
ACF of Squared Standardized Residuals
plot(portfolio_df.garchm11, which = 11)
News-impact curve
plot(portfolio_df.garchm11, which = 12)
Positive and negative shocks of equal magnitude have the same impact on volatility. The U-shaped curve indicates that both positive and negative shocks increase volatility, but the impact diminishes as the magnitude of the shock decreases.
The GARCH-t model
Let’s see whether conditional distribution of the error term can be better described by the t-Student distribution.
Let’s first define a model specification:
spec <- ugarchspec(# variance equation
variance.model = list(model = "sGARCH",
garchOrder = c(1, 1)),
# mean equation
mean.model = list(armaOrder = c(0, 0),
include.mean = TRUE),
# assumed distribution of errors
distribution.model = "std") # std = t-StudentThen, we estimate the model:
portfolio_df.garcht11 <- ugarchfit(spec = spec,
data = portfolio_df$r)Model summary:
portfolio_df.garcht11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.001061 0.000418 2.5398 0.011092
## omega 0.000014 0.000002 7.8966 0.000000
## alpha1 0.092405 0.000656 140.7937 0.000000
## beta1 0.851642 0.019658 43.3229 0.000000
## shape 4.885653 0.687331 7.1082 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.001061 0.000398 2.6667 0.007659
## omega 0.000014 0.000003 5.0925 0.000000
## alpha1 0.092405 0.013962 6.6183 0.000000
## beta1 0.851642 0.015344 55.5031 0.000000
## shape 4.885653 0.600187 8.1402 0.000000
##
## LogLikelihood : 2522.343
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.6570
## Bayes -5.6300
## Shibata -5.6570
## Hannan-Quinn -5.6467
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.297 0.2548
## Lag[2*(p+q)+(p+q)-1][2] 1.624 0.3335
## Lag[4*(p+q)+(p+q)-1][5] 2.290 0.5517
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.006618 0.9352
## Lag[2*(p+q)+(p+q)-1][5] 2.554874 0.4943
## Lag[4*(p+q)+(p+q)-1][9] 3.274714 0.7129
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.1841 0.500 2.000 0.6679
## ARCH Lag[5] 0.5576 1.440 1.667 0.8665
## ARCH Lag[7] 0.6395 2.315 1.543 0.9641
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 21.6399
## Individual Statistics:
## mu 0.3539
## omega 1.6842
## alpha1 0.2489
## beta1 0.4075
## shape 0.3869
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.3693 0.1713
## Negative Sign Bias 0.4521 0.6513
## Positive Sign Bias 0.8065 0.4201
## Joint Effect 3.3422 0.3418
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 12.29 0.8728
## 2 30 19.10 0.9187
## 3 40 30.09 0.8463
## 4 50 53.71 0.2987
##
##
## Elapsed time : 0.06211805Interpration
1.Ljung-Box Tests for Residuals and Squared Residuals: There is no evidence of significant serial correlation in the residuals of your time series model.
2.Arch Test for Residuals: There is no significant ARCH effect present in the residuals of your time series model at the conventional significance level.
3.Nyblom Stability Test: There is evidence of instability in the parameters of your time series model, particularly for “omega” and “beta1”.
4.Sign Bias Test: There is no evidence of systematic sign bias in the residuals of your time series model.
5.Adjusted Pearson Goodness-of-Fit Test: There isn’t significant evidence to suggest a lack of fit between the observed and expected frequencies in each group, based on the adjusted Pearson goodness-of-fit test.
Conditional SD (vs |returns|)
plot(portfolio_df.garcht11, which = 3)
There are periods where the volatility is relatively low and stable, suggesting less market turbulence.
ACF of Standardized Residuals
plot(portfolio_df.garcht11, which = 10)
ACF of Squared Standardized Residuals
plot(portfolio_df.garcht11, which = 11)
News-impact curve
plot(portfolio_df.garcht11, which = 12)
Positive and negative shocks of equal magnitude have the same impact on volatility. The U-shaped curve indicates that both positive and negative shocks increase volatility, but the impact diminishes as the magnitude of the shock decreases.
The EGARCH in mean-t model
Let’s first define the model specification:
spec <- ugarchspec(# variance equation
variance.model = list(model = "eGARCH",
garchOrder = c(1, 1)),
# mean equation
mean.model = list(armaOrder = c(0, 0),
include.mean = TRUE,
archm = TRUE, archpow = 1),
# assumed distribution of errors
distribution.model = "std") # std = t-StudentNow, we estimate the model:
portfolio_df.egarchmt11 <- ugarchfit(spec = spec,
data = portfolio_df$r)The model summary:
portfolio_df.egarchmt11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.001916 0.001114 -1.719904 0.085450
## archm 0.218793 0.081423 2.687123 0.007207
## omega -0.388420 0.056812 -6.836986 0.000000
## alpha1 -0.000857 0.024814 -0.034522 0.972461
## beta1 0.953938 0.006625 143.996472 0.000000
## gamma1 0.207409 0.038656 5.365490 0.000000
## shape 4.857327 0.753861 6.443268 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.001916 0.000701 -2.732208 0.006291
## archm 0.218793 0.050113 4.365977 0.000013
## omega -0.388420 0.029135 -13.331581 0.000000
## alpha1 -0.000857 0.030138 -0.028423 0.977325
## beta1 0.953938 0.002941 324.388058 0.000000
## gamma1 0.207409 0.047268 4.387916 0.000011
## shape 4.857327 0.763383 6.362897 0.000000
##
## LogLikelihood : 2522.855
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.6536
## Bayes -5.6159
## Shibata -5.6537
## Hannan-Quinn -5.6392
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.197 0.2740
## Lag[2*(p+q)+(p+q)-1][2] 1.820 0.2945
## Lag[4*(p+q)+(p+q)-1][5] 2.748 0.4551
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.004974 0.94377
## Lag[2*(p+q)+(p+q)-1][5] 5.869845 0.09651
## Lag[4*(p+q)+(p+q)-1][9] 7.008826 0.19842
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.1570 0.500 2.000 0.6920
## ARCH Lag[5] 0.6140 1.440 1.667 0.8495
## ARCH Lag[7] 0.7192 2.315 1.543 0.9545
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.3704
## Individual Statistics:
## mu 0.4842
## archm 0.4970
## omega 0.1803
## alpha1 0.1417
## beta1 0.1916
## gamma1 0.1379
## shape 0.2291
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.9857 0.3246
## Negative Sign Bias 0.4686 0.6395
## Positive Sign Bias 0.7294 0.4660
## Joint Effect 1.9941 0.5736
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 15.93 0.6618
## 2 30 30.63 0.3831
## 3 40 37.55 0.5360
## 4 50 61.57 0.1072
##
##
## Elapsed time : 0.1682701Interpration
1.Ljung-Box Tests for Residuals and Squared Residuals: There is no significant serial correlation present in the residuals or their squares based on these tests.
2.Arch Test for Residuals: There is not enough evidence to reject the null hypothesis of no conditional heteroskedasticity at any of the tested lag lengths. Therefore, based on these results, the data do not exhibit significant conditional heteroskedasticity.
3.Nyblom Stability Test: There is no evidence to suggest parameter instability in your model.
4.Sign Bias Test: there’s no significant evidence to suggest a systematic bias in the predictions of your model.
5.Adjusted Pearson Goodness-of-Fit Test: There’s no strong evidence to suggest significant deviations between the observed data and the specified theoretical distribution for most groups.
Conditional SD (vs |returns|)
plot(portfolio_df.garcht11, which = 3)
There are periods where the volatility is relatively low and stable, suggesting less market turbulence.
ACF of Standardized Residuals
plot(portfolio_df.garcht11, which = 10)
ACF of Squared Standardized Residuals
plot(portfolio_df.garcht11, which = 11)
News-impact curve
plot(portfolio_df.garcht11, which = 12)
Positive and negative shocks of equal magnitude have the same impact on volatility. The U-shaped curve indicates that both positive and negative shocks increase volatility, but the impact diminishes as the magnitude of the shock decreases.
Now, let’s compare information criteria for all models:
compare_ICs_ugarchfit(c("portfolio_df.garch11",
"portfolio_df.egarch11",
"portfolio_df.garchm11",
"portfolio_df.garcht11",
"portfolio_df.egarchmt11"))## $ICs
## Akaike Bayes Shibata Hannan.Quinn model
## 1 -5.549567 -5.528033 -5.549607 -5.541336 portfolio_df.garch11
## 2 -5.547386 -5.520469 -5.547449 -5.537098 portfolio_df.egarch11
## 3 -5.549036 -5.522119 -5.549099 -5.538748 portfolio_df.garchm11
## 4 -5.656951 -5.630034 -5.657014 -5.646663 portfolio_df.garcht11
## 5 -5.653606 -5.615923 -5.653729 -5.639203 portfolio_df.egarchmt11
##
## $which.min
## Akaike Bayes Shibata Hannan.Quinn
## 4 4 4 4Calculation of VaR
VaR in the IN-SAMPLE period for the GARCH-t model
Standardization of returns.
portfolio_df$rstd <- (portfolio_df$r - mean(portfolio_df$r, na.rm = T)) /
sd(portfolio_df$r ,na.rm = T)
tail(portfolio_df)## date r stan rstd
## 885 2022-12-22 -0.013659920 -0.86859005 -0.86859005
## 886 2022-12-23 0.002053295 0.06072186 0.06072186
## 887 2022-12-27 0.003362720 0.13816399 0.13816399
## 888 2022-12-28 -0.011738050 -0.75492673 -0.75492673
## 889 2022-12-29 0.015381272 0.84896594 0.84896594
## 890 2022-12-30 -0.006366028 -0.43721426 -0.437214261% empirical quantile
q01 <- quantile(portfolio_df$rstd, 0.01, na.rm = T)
q01## 1%
## -2.631463For comparison: 1% quantile of standard normal distribution
qnorm(0.01, 0, 1)## [1] -2.326348str(portfolio_df.garcht11)## Formal class 'uGARCHfit' [package "rugarch"] with 2 slots
## ..@ fit :List of 27
## .. ..$ hessian : num [1:5, 1:5] 5736246 5827313 3135 1829 119 ...
## .. ..$ cvar : num [1:5, 1:5] 1.75e-07 -6.59e-12 -4.52e-08 1.57e-07 -1.04e-05 ...
## .. ..$ var : num [1:890] 0.000286 0.00034 0.000327 0.000292 0.000273 ...
## .. ..$ sigma : num [1:890] 0.0169 0.0184 0.0181 0.0171 0.0165 ...
## .. ..$ condH : num NaN
## .. ..$ z : num [1:890] -1.7753 0.8619 -0.0386 0.6134 0.1901 ...
## .. ..$ LLH : num 2522
## .. ..$ log.likelihoods: num [1:890] -1.2 -2.61 -3.3 -3 -3.36 ...
## .. ..$ residuals : num [1:890] -0.03 0.015899 -0.000698 0.010489 0.003142 ...
## .. ..$ coef : Named num [1:5] 1.06e-03 1.39e-05 9.24e-02 8.52e-01 4.89
## .. .. ..- attr(*, "names")= chr [1:5] "mu" "omega" "alpha1" "beta1" ...
## .. ..$ robust.cvar : num [1:5, 1:5] 1.58e-07 6.93e-11 -3.95e-07 -2.04e-07 -3.16e-06 ...
## .. ..$ A : num [1:5, 1:5] 6445.22 6547.543 3.523 2.055 0.134 ...
## .. ..$ B : num [1:5, 1:5] 5.84e+03 -8.27e+04 2.91 -5.50 -8.96e-04 ...
## .. ..$ scores : num [1:890, 1:5] 102.16 -77.48 7.14 -63.8 -24.03 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : NULL
## .. .. .. ..$ : chr [1:5] "mu" "omega" "alpha1" "beta1" ...
## .. ..$ se.coef : num [1:5] 4.18e-04 1.76e-06 6.56e-04 1.97e-02 6.87e-01
## .. ..$ tval : Named num [1:5] 2.54 7.9 140.79 43.32 7.11
## .. .. ..- attr(*, "names")= chr [1:5] "mu" "omega" "alpha1" "beta1" ...
## .. ..$ matcoef : num [1:5, 1:4] 1.06e-03 1.39e-05 9.24e-02 8.52e-01 4.89 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:5] "mu" "omega" "alpha1" "beta1" ...
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ robust.se.coef : num [1:5] 3.98e-04 2.72e-06 1.40e-02 1.53e-02 6.00e-01
## .. ..$ robust.tval : Named num [1:5] 2.67 5.09 6.62 55.5 8.14
## .. .. ..- attr(*, "names")= chr [1:5] "mu" "omega" "alpha1" "beta1" ...
## .. ..$ robust.matcoef : num [1:5, 1:4] 1.06e-03 1.39e-05 9.24e-02 8.52e-01 4.89 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:5] "mu" "omega" "alpha1" "beta1" ...
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ fitted.values : num [1:890] 0.00106 0.00106 0.00106 0.00106 0.00106 ...
## .. ..$ convergence : num 0
## .. ..$ kappa : num 1
## .. ..$ persistence : num 0.944
## .. ..$ timer : 'difftime' num 0.0621180534362793
## .. .. ..- attr(*, "units")= chr "secs"
## .. ..$ ipars : num [1:19, 1:6] 0.00106 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ solver :List of 2
## .. .. ..$ sol :List of 10
## .. .. .. ..$ pars : Named num [1:5] 1.06e-03 1.39e-05 9.24e-02 8.52e-01 4.89
## .. .. .. .. ..- attr(*, "names")= chr [1:5] "mu" "omega" "alpha1" "beta1" ...
## .. .. .. ..$ convergence: num 0
## .. .. .. ..$ values : num [1:3] -2395 -2522 -2522
## .. .. .. ..$ lagrange : num [1, 1] 0.000247
## .. .. .. ..$ hessian : num [1:6, 1:6] 0.337 10.671 7498.048 -0.862 -1.437 ...
## .. .. .. ..$ ineqx0 : Named num 0.944
## .. .. .. .. ..- attr(*, "names")= chr ""
## .. .. .. ..$ nfuneval : num 261
## .. .. .. ..$ outer.iter : num 2
## .. .. .. ..$ elapsed : 'difftime' num 0.0615501403808594
## .. .. .. .. ..- attr(*, "units")= chr "secs"
## .. .. .. ..$ vscale : num [1:7] 1 1 1 1 1 1 1
## .. .. ..$ hess: NULL
## ..@ model:List of 11
## .. ..$ modelinc : Named num [1:22] 1 0 0 0 0 0 1 1 1 0 ...
## .. .. ..- attr(*, "names")= chr [1:22] "mu" "ar" "ma" "arfima" ...
## .. ..$ modeldesc :List of 3
## .. .. ..$ distribution: chr "std"
## .. .. ..$ distno : int 3
## .. .. ..$ vmodel : chr "sGARCH"
## .. ..$ modeldata :List of 4
## .. .. ..$ T : int 890
## .. .. ..$ data : num [1:890] -0.028939 0.01696 0.000363 0.01155 0.004203 ...
## .. .. ..$ index : POSIXct[1:890], format: "1970-01-02" ...
## .. .. ..$ period: 'difftime' num 1
## .. .. .. ..- attr(*, "units")= chr "days"
## .. ..$ pars : num [1:19, 1:6] 0.00106 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ start.pars: NULL
## .. ..$ fixed.pars: NULL
## .. ..$ maxOrder : num 1
## .. ..$ pos.matrix: num [1:21, 1:3] 1 0 0 0 0 0 2 3 4 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:21] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:3] "start" "stop" "include"
## .. ..$ fmodel : NULL
## .. ..$ pidx : num [1:19, 1:2] 1 2 3 4 5 6 7 8 9 10 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:2] "begin" "end"
## .. ..$ n.start : num 0head(portfolio_df.garcht11@fit$sigma)## [1] 0.01689897 0.01844567 0.01808292 0.01709957 0.01652433 0.01572661Calculating value-at-risk (VaR) the GARCH-t model.
portfolio_df$var <- q01*portfolio_df.garcht11@fit$sigma
tail(portfolio_df)## date r stan rstd var
## 885 2022-12-22 -0.013659920 -0.86859005 -0.86859005 -0.04057937
## 886 2022-12-23 0.002053295 0.06072186 0.06072186 -0.04046103
## 887 2022-12-27 0.003362720 0.13816399 0.13816399 -0.03861211
## 888 2022-12-28 -0.011738050 -0.75492673 -0.75492673 -0.03700195
## 889 2022-12-29 0.015381272 0.84896594 0.84896594 -0.03697144
## 890 2022-12-30 -0.006366028 -0.43721426 -0.43721426 -0.03730095Plot of returns vs value-at-risk.
plot(portfolio_df$date,portfolio_df$r, col="red",
lwd=1,type="l", ylim= c(-0.1, 0.1))
abline(h = 0, lty = 2)
lines(portfolio_df$date,portfolio_df$var,col="green", type="l")
In how many days losses were higher than the assumed value-at-risk?
sum(portfolio_df$r < portfolio_df$var)/ length(portfolio_df$var)## [1] 0.01348315VaR in the OUT-OF-SAMPLE period fot the GARCH-t model
Plot of conditional standard deviation and its on-day ahead prediction.
plot(ugarchforecast(portfolio_df.garcht11, na.ahead=1), which=3)
Plot of conditional standard deviation forecasts in the long run.
plot(ugarchforecast(portfolio_df.garcht11, n.ahead = 200), which=3)
We can combine them with the in-sample estimation of conditional standard deviation
sigma.forecast.longrun <- ugarchforecast(portfolio_df.garcht11, n.ahead = 500)
unconditional_sigma <-
sqrt(
portfolio_df.garch11@model$pars["omega", 1] /
(1 -
portfolio_df.garcht11@model$pars["alpha1", 1] -
portfolio_df.garcht11@model$pars["beta1", 1]))
plot(
c(as.numeric(portfolio_df.garcht11@fit$sigma),
as.numeric(sigma.forecast.longrun@forecast$sigmaFor)),
type = "l",
ylab = "sigma")
abline(h = unconditional_sigma, col = "red")
Yet a better idea is to annualize all of these values.
plot(
c(as.numeric(portfolio_df.garcht11@fit$sigma * sqrt(252)),
as.numeric(sigma.forecast.longrun@forecast$sigmaFor * sqrt(252))),
type = "l",
ylab = "sigma annualized")
abline(h = unconditional_sigma * sqrt(252), col = "red")
Now, the loop below calculates predictions of VaR for the whole OUT-OF-SAMPLE period
portfolio_df1 <-
portfolio_df1 %>%
mutate(obs = row_number())
start <- portfolio_df1$obs[portfolio_df1$date == as.Date("2023-01-03")]
finish <- portfolio_df1$obs[portfolio_df1$date == as.Date("2024-05-30")]
portfolio_df2 <- portfolio_df1[start:finish, ]
VaR <- rep(NA, times = finish - start + 1)
mu <- rep(NA, times = finish - start + 1)
omega <- rep(NA, times = finish - start + 1)
alpha1 <- rep(NA, times = finish - start + 1)
beta1 <- rep(NA, times = finish - start + 1)Calculation lasts for ~90-120 seconds:
time1 <- Sys.time()
for (k in start:finish) {
tmp.data <- portfolio_df1[portfolio_df1$obs <= (k - 1), ]
tmp.data <- tmp.data[as.Date("2019-01-01") <= tmp.data$date, ]
tmp.data$rstd <-
(tmp.data$r - mean(tmp.data$r, na.rm = T)) / sd(tmp.data$r, na.rm = T)
q01 <- quantile(tmp.data$rstd, 0.01, na.rm = T)
spec <-
ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = T),
distribution.model = "std")
tmp.garch11 <- ugarchfit(spec = spec, data = na.omit(tmp.data$r))
sigma.forecast <- ugarchforecast(tmp.garch11, n.ahead = 1)
sigma.forecast2 <- sigma.forecast@forecast$sigmaFor[1, 1]
VaR[k - start + 1] <- q01 * sigma.forecast2
mu[k - start + 1] <- tmp.garch11@fit$coef[1]
omega[k - start + 1] <- tmp.garch11@fit$coef[2]
alpha1[k - start + 1] <- tmp.garch11@fit$coef[3]
beta1[k - start + 1] <- tmp.garch11@fit$coef[4]
}
time2 <- Sys.time()
time2 - time1## Time difference of 31.10518 secsAdding VaR values to the data.frame objects.
portfolio_df2$Var <- VaR
head(portfolio_df2$Var)## [1] -0.03628062 -0.03488198 -0.03506423 -0.03469775 -0.03537544 -0.03432390Plot of returns vs. VaR in the OUT-OF-SAMPLE period
plot(portfolio_df2$date, portfolio_df2$r, col = "red", lwd = 1, type = 'l',
ylim = c(-0.20, 0.20))
abline(h = 0, lty = 2)
lines(portfolio_df2$date, portfolio_df2$Var, type = 'l', col = "green")
In how many days losses were higher than the assumed VaR?
sum(portfolio_df2$r < portfolio_df2$Var) / length(portfolio_df2$Var)## [1] 0VaR in the IN-SAMPLE period for the GARCH-mean-t model
str(portfolio_df.egarchmt11)## Formal class 'uGARCHfit' [package "rugarch"] with 2 slots
## ..@ fit :List of 27
## .. ..$ hessian : num [1:7, 1:7] 5860725 87395 84383 -20633 -620726 ...
## .. ..$ cvar : num [1:7, 1:7] -1.24e-06 9.80e-05 -1.73e-05 1.01e-07 -1.89e-06 ...
## .. ..$ var : num [1:890] 0.000294 0.000362 0.000357 0.000306 0.00029 ...
## .. ..$ sigma : num [1:890] 0.0172 0.019 0.0189 0.0175 0.017 ...
## .. ..$ condH : num NaN
## .. ..$ z : num [1:890] -1.7942 0.773 -0.0981 0.5515 0.1407 ...
## .. ..$ LLH : num 2523
## .. ..$ log.likelihoods: num [1:890] -1.15 -2.7 -3.25 -3.05 -3.35 ...
## .. ..$ residuals : num [1:890] -0.03078 0.01471 -0.00185 0.00964 0.0024 ...
## .. ..$ coef : Named num [1:7] -0.001916 0.218793 -0.38842 -0.000857 0.953938 ...
## .. .. ..- attr(*, "names")= chr [1:7] "mu" "archm" "omega" "alpha1" ...
## .. ..$ robust.cvar : num [1:7, 1:7] 4.92e-07 -2.75e-05 -8.16e-07 1.70e-06 6.86e-09 ...
## .. ..$ A : num [1:7, 1:7] 6585.1 98.2 94.8 -23.2 -697.4 ...
## .. ..$ B : num [1:7, 1:7] 5820 79.3 56.2 15.4 -458.5 ...
## .. ..$ scores : num [1:890, 1:7] 118.07 -68.31 3.46 -58.71 -23.1 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : NULL
## .. .. .. ..$ : chr [1:7] "mu" "archm" "omega" "alpha1" ...
## .. ..$ se.coef : num [1:7] 0.00111 0.08142 0.05681 0.02481 0.00662 ...
## .. ..$ tval : Named num [1:7] -1.7199 2.6871 -6.837 -0.0345 143.9965 ...
## .. .. ..- attr(*, "names")= chr [1:7] "mu" "archm" "omega" "alpha1" ...
## .. ..$ matcoef : num [1:7, 1:4] -0.001916 0.218793 -0.38842 -0.000857 0.953938 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:7] "mu" "archm" "omega" "alpha1" ...
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ robust.se.coef : num [1:7] 0.000701 0.050113 0.029135 0.030138 0.002941 ...
## .. ..$ robust.tval : Named num [1:7] -2.7322 4.366 -13.3316 -0.0284 324.3881 ...
## .. .. ..- attr(*, "names")= chr [1:7] "mu" "archm" "omega" "alpha1" ...
## .. ..$ robust.matcoef : num [1:7, 1:4] -0.001916 0.218793 -0.38842 -0.000857 0.953938 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:7] "mu" "archm" "omega" "alpha1" ...
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ fitted.values : num [1:890] 0.00184 0.00225 0.00222 0.00191 0.00181 ...
## .. ..$ convergence : num 0
## .. ..$ kappa : num 0.732
## .. ..$ persistence : num 0.954
## .. ..$ timer : 'difftime' num 0.168270111083984
## .. .. ..- attr(*, "units")= chr "secs"
## .. ..$ ipars : num [1:19, 1:6] -0.00192 0 0 0 0.21879 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ solver :List of 2
## .. .. ..$ sol :List of 10
## .. .. .. ..$ pars : Named num [1:7] -0.001916 0.218793 -0.38842 -0.000857 0.953938 ...
## .. .. .. .. ..- attr(*, "names")= chr [1:7] "mu" "archm" "omega" "alpha1" ...
## .. .. .. ..$ convergence: num 0
## .. .. .. ..$ values : num [1:3] 294 -2523 -2523
## .. .. .. ..$ lagrange : num [1, 1] 2.8e-05
## .. .. .. ..$ hessian : num [1:8, 1:8] 0.405 7.943 0.285 2.192 0.494 ...
## .. .. .. ..$ ineqx0 : Named num 1.05
## .. .. .. .. ..- attr(*, "names")= chr ""
## .. .. .. ..$ nfuneval : num 224
## .. .. .. ..$ outer.iter : num 2
## .. .. .. ..$ elapsed : 'difftime' num 0.0862560272216797
## .. .. .. .. ..- attr(*, "units")= chr "secs"
## .. .. .. ..$ vscale : num [1:9] 1 1 1 1 1 1 1 1 1
## .. .. ..$ hess: NULL
## ..@ model:List of 11
## .. ..$ modelinc : Named num [1:22] 1 0 0 0 1 0 1 1 1 1 ...
## .. .. ..- attr(*, "names")= chr [1:22] "mu" "ar" "ma" "arfima" ...
## .. ..$ modeldesc :List of 3
## .. .. ..$ distribution: chr "std"
## .. .. ..$ distno : int 3
## .. .. ..$ vmodel : chr "eGARCH"
## .. ..$ modeldata :List of 4
## .. .. ..$ T : int 890
## .. .. ..$ data : num [1:890] -0.028939 0.01696 0.000363 0.01155 0.004203 ...
## .. .. ..$ index : POSIXct[1:890], format: "1970-01-02" ...
## .. .. ..$ period: 'difftime' num 1
## .. .. .. ..- attr(*, "units")= chr "days"
## .. ..$ pars : num [1:19, 1:6] -0.00192 0 0 0 0.21879 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ start.pars: NULL
## .. ..$ fixed.pars: NULL
## .. ..$ maxOrder : num 1
## .. ..$ pos.matrix: num [1:21, 1:3] 1 0 0 0 2 0 3 4 5 6 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:21] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:3] "start" "stop" "include"
## .. ..$ fmodel : NULL
## .. ..$ pidx : num [1:19, 1:2] 1 2 3 4 5 6 7 8 9 10 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:2] "begin" "end"
## .. ..$ n.start : num 0head(portfolio_df.egarchmt11@fit$sigma)## [1] 0.01715327 0.01903247 0.01888400 0.01748296 0.01702073 0.01590208Calculating value-at-risk (VaR) the GARCH-mean-t model.
portfolio_df$varmt <- q01*portfolio_df.egarchmt11@fit$sigma
tail(portfolio_df)## date r stan rstd var varmt
## 885 2022-12-22 -0.013659920 -0.86859005 -0.86859005 -0.04057937 -0.04109564
## 886 2022-12-23 0.002053295 0.06072186 0.06072186 -0.04046103 -0.04207721
## 887 2022-12-27 0.003362720 0.13816399 0.13816399 -0.03861211 -0.03899856
## 888 2022-12-28 -0.011738050 -0.75492673 -0.75492673 -0.03700195 -0.03668207
## 889 2022-12-29 0.015381272 0.84896594 0.84896594 -0.03697144 -0.03756782
## 890 2022-12-30 -0.006366028 -0.43721426 -0.43721426 -0.03730095 -0.03869597Plot of returns vs value-at-risk.
plot(portfolio_df$date,portfolio_df$r, col="red",
lwd=1,type="l", ylim= c(-0.1, 0.1))
abline(h = 0, lty = 2)
lines(portfolio_df$date,portfolio_df$varmt,col="green", type="l")
In how many days losses were higher than the assumed value-at-risk?
sum(portfolio_df$r < portfolio_df$varmt)/ length(portfolio_df$varmt)## [1] 0.01348315VaR in the OUT-OF-SAMPLE period for the GARCH-mean-t model
Plot of conditional standard deviation and its on-day ahead prediction.
plot(ugarchforecast(portfolio_df.egarchmt11, na.ahead=1), which=3)
Plot of conditional standard deviation forecasts in the long run.
plot(ugarchforecast(portfolio_df.egarchmt11, n.ahead = 200), which=3)
We can combine them with the in-sample estimation of conditional standard deviation
sigma.forecast.longrun <- ugarchforecast(portfolio_df.egarchmt11, n.ahead = 500)
unconditional_sigma <-
sqrt(
portfolio_df.egarchmt11@model$pars["omega", 1] /
(1 -
portfolio_df.egarchmt11@model$pars["alpha1", 1] -
portfolio_df.egarchmt11@model$pars["beta1", 1]))## Warning in sqrt(portfolio_df.egarchmt11@model$pars["omega", 1]/(1 -
## portfolio_df.egarchmt11@model$pars["alpha1", : NaNs producedplot(
c(as.numeric(portfolio_df.egarchmt11@fit$sigma),
as.numeric(sigma.forecast.longrun@forecast$sigmaFor)),
type = "l",
ylab = "sigma")
abline(h = unconditional_sigma, col = "red")
Yet a better idea is to annualize all of these values.
plot(
c(as.numeric(portfolio_df.egarchmt11@fit$sigma * sqrt(252)),
as.numeric(sigma.forecast.longrun@forecast$sigmaFor * sqrt(252))),
type = "l",
ylab = "sigma annualized")
abline(h = unconditional_sigma * sqrt(252), col = "red")
Now, the loop below calculates predictions of VaR for the whole OUT-OF-SAMPLE period
portfolio_df1 <-
portfolio_df1 %>%
mutate(obs = row_number())
start <- portfolio_df1$obs[portfolio_df1$date == as.Date("2023-01-03")]
finish <- portfolio_df1$obs[portfolio_df1$date == as.Date("2024-05-30")]
portfolio_df2 <- portfolio_df1[start:finish, ]
VaR <- rep(NA, times = finish - start + 1)
mu <- rep(NA, times = finish - start + 1)
omega <- rep(NA, times = finish - start + 1)
alpha1 <- rep(NA, times = finish - start + 1)
beta1 <- rep(NA, times = finish - start + 1)Calculation lasts for ~90-120 seconds:
time1 <- Sys.time()
for (k in start:finish) {
tmp.data <- portfolio_df1[portfolio_df1$obs <= (k - 1), ]
tmp.data <- tmp.data[as.Date("2019-01-01") <= tmp.data$date, ]
tmp.data$rstd <-
(tmp.data$r - mean(tmp.data$r, na.rm = T)) / sd(tmp.data$r, na.rm = T)
q01 <- quantile(tmp.data$rstd, 0.01, na.rm = T)
spec <-
ugarchspec(variance.model = list(model = "eGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = T),
distribution.model = "std")
tmp.garch11 <- ugarchfit(spec = spec, data = na.omit(tmp.data$r))
sigma.forecast <- ugarchforecast(tmp.garch11, n.ahead = 1)
sigma.forecast2 <- sigma.forecast@forecast$sigmaFor[1, 1]
VaR[k - start + 1] <- q01 * sigma.forecast2
mu[k - start + 1] <- tmp.garch11@fit$coef[1]
omega[k - start + 1] <- tmp.garch11@fit$coef[2]
alpha1[k - start + 1] <- tmp.garch11@fit$coef[3]
beta1[k - start + 1] <- tmp.garch11@fit$coef[4]
}
time2 <- Sys.time()
time2 - time1## Time difference of 54.66239 secsAdding VaR values to the data.frame objects.
portfolio_df2$Var <- VaR
head(portfolio_df2$Var)## [1] -0.03771902 -0.03527382 -0.03613666 -0.03618297 -0.03748307 -0.03609196Plot of returns vs. VaR in the OUT-OF-SAMPLE period
plot(portfolio_df2$date, portfolio_df2$r, col = "red", lwd = 1, type = 'l',
ylim = c(-0.20, 0.20))
abline(h = 0, lty = 2)
lines(portfolio_df2$date, portfolio_df2$Var, type = 'l', col = "green")
In how many days losses were higher than the assumed VaR?
sum(portfolio_df2$r < portfolio_df2$Var) / length(portfolio_df2$Var)## [1] 0